Novel similarity measures under complex pythagorean fuzzy soft matrices and their application in decision making problems

Complex fuzzy soft matrices play a crucial role in various applications, including decision-making, pattern recognition, signals processing, and image processing. The main objective of this study is to introduce the unique notions of complex Pythagorean fuzzy soft matrices (CPFSMs), which provide more flexibility and accuracy in modelling uncertainty. CPFSMs incorporate Pythagorean fuzzy soft matrices, allowing for more sophisticated uncertainty modeling. The key findings of CPFSMs, specific instances, and certain fundamental set-theoretic operations and principles were covered. A set of new distance metrics between two CPFSMs has been defined. In the context of complex Pythagorean fuzzy soft sets and complex Pythagorean fuzzy soft matrices, we created a CPFS decision-making technique. Moreover, the application’s numerical example and comparison analysis have been effectively demonstrated. Thus, by integrating the concepts of Pythagorean fuzzy sets, soft matrices, and complex numbers, CPFSMs provide a robust framework with membership and non-membership degrees for complex decision-making modeling and analyzing uncertain data.


Preliminaries
We will discuss here the basic set-theoretic operations and laws of CPFS sets and matrices and also discuss particular examples of these operations and laws.Definition 2.1 1 Let X be an initial universe set and Ŵ be a set of parameters.Let P(X) denote the power set of X.Consider a nonempty set τ , τ ⊂ Ŵ .A pair(̥, Ŵ) is called a soft set over X, where ̥ is a mapping given by ̥ : τ → P(X).Definition 2.2 15 Let ̥(X) denotes the set of all fuzzy sets of X.A pair (̥, Ŵ) is called a fuzzy soft set over ̥(X) , where ̥ is a mapping given by ̥ : τ → P(̥(X)).

Complex Pythagorean fuzzy soft matrix theory
In this section, we introduce a novel concept: the introduction of a CPFSM.It's notable to mention that a CPFSM surpasses a complex fuzzy soft matrix in generality.This is because each entry of the matrix incorporates both the degree of membership function and non-membership function, offering a more comprehensive selection in decision-making problems.

Why do we need this new model?
Although existing fuzzy soft matrices are highly effective in dealing with ambiguous information, their limitations highlight the need for a superior model that can serve as a powerful tool in dealing with ambiguous information.The principle of fuzzy soft matrices has been utilized in separated areas, but the principle of fuzzy soft matrices has limited applications due to its structure.Because if a person faces information in the form of TG and falsity grade (FG), then the fuzzy soft matrices principle has failed in certain actual life troubles.To overcome this difficulty, we introduce the concepts of CPFSMs.False grades in CPFSMs play a vital role in defining the features of CPFSMs.This term distinguishes a CPFSM from all other soft matrices in the literature.The CPFSMs discuss two-dimensional phenomena in the form of truth and false grades, making them superior to handling ambiguous and intuitive information prevalent in time-periodic phenomena.They provide a powerful and flexible framework for modeling and analyzing complex decision-making problems, which are increasingly prevalent in various fields, including management, economics, engineering, and healthcare.Traditional approaches often struggle to handle uncertainty, ambiguity, and multi-criteria decision-making challenges, leading to inaccurate or suboptimal decisions.CPFSMs address these limitations by integrating Pythagorean fuzzy sets, soft matrices, and complex numbers, enabling the simultaneous consideration of multiple criteria, uncertainty levels, and complex relationships.This allows for more accurate and informed decision-making outcomes, making CPFSMs a crucial tool for tackling complex real-world problems and enabling better decision-making in uncertain and dynamic environments.

Example 3.3 Assume that
Then and for all ŝ and ȗ.
Definition 3.7 Let [L] ŝ×ȗ and [ß] ŝ×ȗ be CPFSMs.Then, the difference between these two CPFSMs is defined by Definition 3.8 Let [L] ŝ×ȗ and [ß] ŝ×ȗ be CPFSMs.Then, the symmetric difference of these two CPFSMs denoted by [L] ŝ×ȗ [ß] ŝ×ȗ and is defined as: Definition 3.9 If we interchange the operation union ' ∪ ' and intersection ' ∩ ' to each other in any result for a CPFSM, we obtain the true result.Therefore, if any result is derivable from these operations, so is the true result is obtained by interchanging union by intersection and intersection by union.This is called the duality principle.

Example 3.4 Let
Thus, the commutative law holds concerning the union.Now, to check the duality principle, we have to change the union by an intersection as follows: Thus, the commutative law again holds.This means that by interchanging the We get the same derivable result by union by intersection and intersection by union.

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
So by the transitive property, we have that So by the transitive property, we have that Vol Proof To prove (i), we take two cases here.
From (4) and ( 5), we have 6) and ( 7), we have Thus, in all cases, the commutative law of union holds.(ii).By duality principle (ii) is also hold.
Proof To prove (i), we take three cases here.
Proof To prove (i), we take three cases here.

Distance measures of complex Pythagorean fuzzy soft matrices
In this section, we will define a novel distance measure for CPFSMs.Distance measures of CPFSMs have farreaching significance and applications in various fields, including decision-making, uncertainty analysis, and data analytics.These measures enable the assessment of similarity and dissimilarity between CPFSMs, which is crucial for evaluating alternative solutions, identifying patterns, and optimizing complex systems.In decisionmaking, distance measures of CPFSMs help select the most suitable option by quantifying the proximity between alternatives.In uncertainty analysis, they facilitate the evaluation of uncertainty levels and risk assessment.Additionally, distance measures of CPFSMs have applications in image processing, natural language processing, and recommender systems, which aid in image segmentation, text classification, and personalized recommendations.Furthermore, they are useful in social network analysis, clustering, and classification, enabling the identification of influential nodes, clusters, and patterns.Overall, distance measures of CPFSMs provide a powerful tool for analyzing and understanding complex data, leading to more informed decisions and improved outcomes.
We introduce the distance measure ∐ as: Note that the distance measure ∐ plays a key role in the remainder of this paper. Vol.:(0123456789) Definition 5.2 A weighted distance measure of CPFSMs is a function ∐ w : M ŝ×ȗ × M ŝ×ȗ → [0, 1] with the properties: for any We introduce the weighted distance measure ∐ w as: where w t is a weighted vector corresponding to parameter.

Theorem 5.3 The function ∐ defined by the equality (21) is a distance measure of CPFSMs on U.
Proof It is easy to prove.

Applications of complex Pythagorean fuzzy soft matrix in decision-making
CPFSMs have many applications in DM problems due to their ability to handle uncertain, complex, and vague information.CPFSMs are applied in MCDM to evaluate alternatives based on multiple criteria.By incorporating degrees of membership and non-membership, CPFSMs can effectively model the uncertainty and ambiguity inherent in DM processes involving various criteria.CPFSMs provide a robust framework for solving, analyzing, and modeling complex DM problems, thereby enhancing the reliability and quality of decisions made in practical applications.
We discuss the subsequent definitions of a DM matrix, considering the notions of a CPFSM.

Normalization process
In the normalization process, we normalize the TG and FG of the CPFS DM matrix as follows: This normalization process transforms the CPFS DM matrix χ into CPFS DM matrix Ŵ: Definition 6.2 Assume that and thus is called positive CPFS DM matrix for scheme i .Definition 6.3 Assume that and thus is called negative CPFS DM matrix for scheme i .

Graphical representation of algorithm
The graphical representation of an algorithm is given below in Fig. 1:

Algorithm
Below are the steps outlining our algorithm for addressing a standard DM problem utilizing the CPFSM alongside positive and negative CPFS DM matrices.
Step 1. Construct the CPFS DM matrix χ corresponding to CPFS sets.
Normalize the CPFS DM matrices χ i into CPFS DM matrix Ŵ i .
Compute the CPFSMs of each scheme i .
Compute each scheme's positive and negative CPFSMs i .
Find the distance measures between positive and negative CPFS DM matrices.
Rank the scheme based on distance measures of CPFSMs.The scheme will be better if the distance measures between positive and The scheme's adverse CPFS decisionmaking matrices are more significant than the other distances.
Step 1.The CPFS DM matrix χ during the three years is given below: www.nature.com/scientificreports/Remark 7.1 27 The proposed distance measure of CPFSMs reduce to the the environment of IFSSs if we consider the imaginary part as zero in truth grade and false grade and take 0 Equation ( 22) represents the distance measure in the environment of IFSSs.
Remark 7.2 20 The distance measure of CPFSMs reduces to the distance the measure of CFSMs if we considered the false grade of CPFSSs as zero, then we have   www.nature.com/scientificreports/ The distance measures between positive and negative CFS decision-making matrices of the schemes 1 , 2 , and 3 are: Therefore, we conclude that the scheme 3 is a better choice for investment.problems, particularly in dynamic and uncertain environments.

Conclusion
Complex fuzzy soft matrices and CPFSMs are both mathematical tools used for handling uncertainty and complexity in decision-making problems.But CPFSMs handle membership and non-membership functions, which can capture more complex and nuanced uncertainty.The novel notions of CPFSMs were defined in this paper.
Basic CPFSM laws and attributes were explored.The set-theoretic operations, particular examples, and main results on CPFSMs were considered.A few novel distance metrics between two CPFSMs were presented by us.
In the context of CPFSSs and CPFSMs, we developed a CPFS DM technique.We talked about how CPFSMs have been used to decision-making difficulties, with encouraging findings that have produced more precise and knowledgeable decision-making outcomes.Additionally, the comparison analysis of CFSMs and CPFSMs was presented.CPFSMs offer more advance and flexible tools for handling complex uncertainty, making them a better choice for intricate decision-making problems, while CFSMs remain suitable for relatively simpler uncertainty scenarios.
In the future, the researchers intend to expand upon the discussed work to encompass interval-valued CPF-SMs, complex neutrosophic soft matrices, and other related areas, aiming to enhance the quality of their research endeavors.